Mechanics of Strain Gradient Materials

May 14, 2018 — May 18, 2018


  • Albrecht Bertram (University of Magdeburg, Germany)
  • Samuel Forest (Mines ParisTech, Evry, France)

Many materials show a different elastic or plastic behavior if tested on large or on small samples. Such size effects can be ascribed to internal length scales related to the micro-structure. This behavior is for instance observed in indenter tests, torsion, bending, and shear banding. Such effects cannot be simulated by classical constitutive models in which the stresses depend locally on the (dimensionless) strains. One possible extension of the classical models relies on the constitutive inclusion of higher strain gradients, which involve the dimension of length. This allows for an introduction of internal length scales in the constitutive model.
Strain gradient models can further be employed for the regularization of singularities in the classical solutions, which make them also advantageous from a numerical point of view. Moreover, they allow for the conceptually sound introduction of line and point forces into continuum mechanics.

The balance laws and boundary conditions for such materials can be derived by resorting to variational principles. For each primal higher strain gradient a conjugate stress tensor of the same tensorial order has to be introduced, for which a constitutive law is needed. Material modeling becomes a challenging task for such materials.
In the elastic case, all stress tensors may depend on all of the corresponding strain tensors. Even in the linear case, this leads to an enormous amount of material constants. This can, however, be drastically reduced by the assumption of symmetry properties like isotropy or centro-symmetry. In the non-linear case of finite deformations, one has to satisfy invariance principles, which is not trivial.

In the case of plastically deforming metals, primal higher strain gradients are chosen with the aim of describing the behavior of geometrically necessary dislocations.

In the course, the following topics will be considered
• experimental findings for size effects
• balance laws and boundary conditions for strain gradient materials
• the linear theory of elasticity and plasticity of strain gradient materials of arbitrary order
• the application to crystal plasticity
• the finite strain gradient theory for large deformations
• the application of strain gradient models to fracture and damage and to micro-to-macro transitions (media with microstructure).


See also